"The (RFU) officially would like to inform that it plans to hold talks with these specialists about one of them becoming the head coach of the Russian national team," the union said in a short statement on its website.
Phil Smith, Redknapp's agent, said his client was open to negotiations.
"Absolutely," Smith was quoted as saying by Russian media when asked if Redknapp, sacked by the Spurs last month, would be interested in taking the job.
"Would Harry have a problem coming to Russia? No. He's ready to go and live in any country, especially as big as Russia. I think such offer would definitely interest him if it came, but so far, we haven't heard anything about it."
On Monday, Russian media, quoting RFU and Sports Ministry sources, said Mancini had agreed a four-year deal to become Russia's new coach after officials found a "signed contract" in the safe of former RFU chief Sergei Fursenko.
Fursenko, a close friend of Russian President Vladimir Putin, quit his post last month.
The RFU was forced to reveal the names of the candidates amid intense media speculation about the job, which became vacant following Dutchman Dick Advocaat's departure after Russia's failure to get past the group phase at Euro 2012.
Others on the RFU list are Russian, including Valery Gazzayev, Anatoly Byshovets and Yuri Semin, who coached the national team in the past but were sacked or quit following a string of poor results.
Former CSKA Moscow boss Gazzayev and Valery Nepomnyashchy, who led Cameroon to the 1990 World Cup quarter-finals, called the selection process a joke.
"It's just a circus and I have no comment on it," Gazzayev was quoted as saying by the Sovietsky Sport newspaper, while 68-year-old Nepomnyashchy said: "My name is on the list? Well, it must be a good joke."
Local media and most football experts said, however, that a Russian would be a long shot for the post, with Capello and Guardiola seen as the early front-runners.